src/HOL/Rat.thy
author haftmann
Thu Mar 11 14:39:58 2010 +0100 (2010-03-11)
changeset 35726 059d2f7b979f
parent 35402 115a5a95710a
child 36112 7fa17a225852
permissions -rw-r--r--
tuned prefixes of ac interpretations
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(*  Title:  HOL/Rat.thy
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    Author: Markus Wenzel, TU Muenchen
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*)
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header {* Rational numbers *}
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theory Rat
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imports GCD Archimedean_Field
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uses ("Tools/float_syntax.ML")
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begin
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subsection {* Rational numbers as quotient *}
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subsubsection {* Construction of the type of rational numbers *}
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definition
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  ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
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  "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
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lemma ratrel_iff [simp]:
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  "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
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  by (simp add: ratrel_def)
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lemma refl_on_ratrel: "refl_on {x. snd x \<noteq> 0} ratrel"
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  by (auto simp add: refl_on_def ratrel_def)
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lemma sym_ratrel: "sym ratrel"
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  by (simp add: ratrel_def sym_def)
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lemma trans_ratrel: "trans ratrel"
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proof (rule transI, unfold split_paired_all)
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  fix a b a' b' a'' b'' :: int
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  assume A: "((a, b), (a', b')) \<in> ratrel"
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  assume B: "((a', b'), (a'', b'')) \<in> ratrel"
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  have "b' * (a * b'') = b'' * (a * b')" by simp
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  also from A have "a * b' = a' * b" by auto
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  also have "b'' * (a' * b) = b * (a' * b'')" by simp
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  also from B have "a' * b'' = a'' * b'" by auto
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  also have "b * (a'' * b') = b' * (a'' * b)" by simp
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  finally have "b' * (a * b'') = b' * (a'' * b)" .
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  moreover from B have "b' \<noteq> 0" by auto
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  ultimately have "a * b'' = a'' * b" by simp
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  with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
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qed
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lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
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  by (rule equiv.intro [OF refl_on_ratrel sym_ratrel trans_ratrel])
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lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
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lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
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lemma equiv_ratrel_iff [iff]: 
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  assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
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  shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
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  by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
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typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
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proof
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  have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
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  then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
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qed
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lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
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  by (simp add: Rat_def quotientI)
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declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
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subsubsection {* Representation and basic operations *}
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definition
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  Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
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  "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
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lemma eq_rat:
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  shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
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  and "\<And>a. Fract a 0 = Fract 0 1"
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  and "\<And>a c. Fract 0 a = Fract 0 c"
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  by (simp_all add: Fract_def)
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lemma Rat_cases [case_names Fract, cases type: rat]:
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  assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
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  shows C
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proof -
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  obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
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    by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
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  let ?a = "a div gcd a b"
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  let ?b = "b div gcd a b"
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  from `b \<noteq> 0` have "?b * gcd a b = b"
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    by (simp add: dvd_div_mult_self)
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  with `b \<noteq> 0` have "?b \<noteq> 0" by auto
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  from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
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    by (simp add: eq_rat dvd_div_mult mult_commute [of a])
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  from `b \<noteq> 0` have coprime: "coprime ?a ?b"
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    by (auto intro: div_gcd_coprime_int)
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  show C proof (cases "b > 0")
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    case True
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    note assms
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    moreover note q
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    moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
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    moreover note coprime
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    ultimately show C .
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  next
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    case False
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    note assms
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    moreover from q have "q = Fract (- ?a) (- ?b)" by (simp add: Fract_def)
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    moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
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    moreover from coprime have "coprime (- ?a) (- ?b)" by simp
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    ultimately show C .
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  qed
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qed
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lemma Rat_induct [case_names Fract, induct type: rat]:
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  assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
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  shows "P q"
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  using assms by (cases q) simp
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instantiation rat :: comm_ring_1
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begin
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definition
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  Zero_rat_def: "0 = Fract 0 1"
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definition
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  One_rat_def: "1 = Fract 1 1"
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definition
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  add_rat_def:
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  "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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    ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
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lemma add_rat [simp]:
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  assumes "b \<noteq> 0" and "d \<noteq> 0"
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  shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
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proof -
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  have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
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    respects2 ratrel"
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  by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
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  with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
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qed
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definition
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  minus_rat_def:
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  "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
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lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
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proof -
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  have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
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    by (simp add: congruent_def)
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  then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
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qed
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lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
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  by (cases "b = 0") (simp_all add: eq_rat)
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definition
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  diff_rat_def: "q - r = q + - (r::rat)"
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lemma diff_rat [simp]:
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  assumes "b \<noteq> 0" and "d \<noteq> 0"
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  shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
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  using assms by (simp add: diff_rat_def)
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definition
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  mult_rat_def:
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  "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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    ratrel``{(fst x * fst y, snd x * snd y)})"
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lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
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proof -
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  have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
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    by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
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  then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
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qed
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lemma mult_rat_cancel:
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  assumes "c \<noteq> 0"
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  shows "Fract (c * a) (c * b) = Fract a b"
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proof -
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  from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
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  then show ?thesis by (simp add: mult_rat [symmetric])
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qed
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instance proof
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  fix q r s :: rat show "(q * r) * s = q * (r * s)" 
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    by (cases q, cases r, cases s) (simp add: eq_rat)
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next
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  fix q r :: rat show "q * r = r * q"
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    by (cases q, cases r) (simp add: eq_rat)
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next
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  fix q :: rat show "1 * q = q"
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    by (cases q) (simp add: One_rat_def eq_rat)
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next
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  fix q r s :: rat show "(q + r) + s = q + (r + s)"
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    by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
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next
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  fix q r :: rat show "q + r = r + q"
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    by (cases q, cases r) (simp add: eq_rat)
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next
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  fix q :: rat show "0 + q = q"
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    by (cases q) (simp add: Zero_rat_def eq_rat)
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next
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  fix q :: rat show "- q + q = 0"
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    by (cases q) (simp add: Zero_rat_def eq_rat)
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next
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  fix q r :: rat show "q - r = q + - r"
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    by (cases q, cases r) (simp add: eq_rat)
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next
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  fix q r s :: rat show "(q + r) * s = q * s + r * s"
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    by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
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next
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  show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
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qed
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end
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lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
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  by (induct k) (simp_all add: Zero_rat_def One_rat_def)
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lemma of_int_rat: "of_int k = Fract k 1"
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  by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
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lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
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  by (rule of_nat_rat [symmetric])
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lemma Fract_of_int_eq: "Fract k 1 = of_int k"
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  by (rule of_int_rat [symmetric])
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instantiation rat :: number_ring
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begin
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definition
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  rat_number_of_def: "number_of w = Fract w 1"
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instance proof
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qed (simp add: rat_number_of_def of_int_rat)
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end
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lemma rat_number_collapse:
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  "Fract 0 k = 0"
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  "Fract 1 1 = 1"
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  "Fract (number_of k) 1 = number_of k"
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  "Fract k 0 = 0"
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  by (cases "k = 0")
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    (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
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lemma rat_number_expand [code_unfold]:
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  "0 = Fract 0 1"
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  "1 = Fract 1 1"
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  "number_of k = Fract (number_of k) 1"
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  by (simp_all add: rat_number_collapse)
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lemma iszero_rat [simp]:
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  "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
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  by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
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lemma Rat_cases_nonzero [case_names Fract 0]:
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  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
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  assumes 0: "q = 0 \<Longrightarrow> C"
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  shows C
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proof (cases "q = 0")
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  case True then show C using 0 by auto
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next
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  case False
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  then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
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  moreover with False have "0 \<noteq> Fract a b" by simp
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  with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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  with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
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qed
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subsubsection {* Function @{text normalize} *}
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lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
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proof (cases "b = 0")
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  case True then show ?thesis by (simp add: eq_rat)
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next
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  case False
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  moreover have "b div gcd a b * gcd a b = b"
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    by (rule dvd_div_mult_self) simp
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  ultimately have "b div gcd a b \<noteq> 0" by auto
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  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
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qed
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definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
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  "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
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    else if snd p = 0 then (0, 1)
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    else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
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lemma normalize_crossproduct:
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  assumes "q \<noteq> 0" "s \<noteq> 0"
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  assumes "normalize (p, q) = normalize (r, s)"
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  shows "p * s = r * q"
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proof -
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  have aux: "p * gcd r s = sgn (q * s) * r * gcd p q \<Longrightarrow> q * gcd r s = sgn (q * s) * s * gcd p q \<Longrightarrow> p * s = q * r"
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  proof -
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    assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
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    then have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)" by simp
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    with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
haftmann@35369
   300
  qed
haftmann@35369
   301
  from assms show ?thesis
haftmann@35369
   302
    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
nipkow@33805
   303
qed
nipkow@33805
   304
haftmann@35369
   305
lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   306
  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
haftmann@35369
   307
    split:split_if_asm)
haftmann@35369
   308
haftmann@35369
   309
lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   310
  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
haftmann@35369
   311
    split:split_if_asm)
haftmann@35369
   312
haftmann@35369
   313
lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   314
  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
haftmann@35369
   315
    split:split_if_asm)
haftmann@35369
   316
haftmann@35369
   317
lemma normalize_stable [simp]:
haftmann@35369
   318
  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
haftmann@35369
   319
  by (simp add: normalize_def)
haftmann@35369
   320
haftmann@35369
   321
lemma normalize_denom_zero [simp]:
haftmann@35369
   322
  "normalize (p, 0) = (0, 1)"
haftmann@35369
   323
  by (simp add: normalize_def)
haftmann@35369
   324
haftmann@35369
   325
lemma normalize_negative [simp]:
haftmann@35369
   326
  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
haftmann@35369
   327
  by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
haftmann@35369
   328
haftmann@35369
   329
text{*
haftmann@35369
   330
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
haftmann@35369
   331
*}
haftmann@35369
   332
haftmann@35369
   333
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
haftmann@35369
   334
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
haftmann@35369
   335
                   snd pair > 0 & coprime (fst pair) (snd pair))"
haftmann@35369
   336
haftmann@35369
   337
lemma quotient_of_unique:
haftmann@35369
   338
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   339
proof (cases r)
haftmann@35369
   340
  case (Fract a b)
haftmann@35369
   341
  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))" by auto
haftmann@35369
   342
  then show ?thesis proof (rule ex1I)
haftmann@35369
   343
    fix p
haftmann@35369
   344
    obtain c d :: int where p: "p = (c, d)" by (cases p)
haftmann@35369
   345
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   346
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
haftmann@35369
   347
    have "c = a \<and> d = b"
haftmann@35369
   348
    proof (cases "a = 0")
haftmann@35369
   349
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
haftmann@35369
   350
    next
haftmann@35369
   351
      case False
haftmann@35369
   352
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
haftmann@35369
   353
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
haftmann@35369
   354
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
haftmann@35369
   355
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
haftmann@35369
   356
      from `coprime a b` `coprime c d` have "\<bar>a\<bar> * \<bar>d\<bar> = \<bar>c\<bar> * \<bar>b\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>d\<bar> = \<bar>b\<bar>"
haftmann@35369
   357
        by (simp add: coprime_crossproduct_int)
haftmann@35369
   358
      with `b > 0` `d > 0` have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
haftmann@35369
   359
      then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b" by (simp add: abs_sgn)
haftmann@35369
   360
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
nipkow@33805
   361
    qed
haftmann@35369
   362
    with p show "p = (a, b)" by simp
nipkow@33805
   363
  qed
nipkow@33805
   364
qed
nipkow@33805
   365
haftmann@35369
   366
lemma quotient_of_Fract [code]:
haftmann@35369
   367
  "quotient_of (Fract a b) = normalize (a, b)"
haftmann@35369
   368
proof -
haftmann@35369
   369
  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
haftmann@35369
   370
    by (rule sym) (auto intro: normalize_eq)
haftmann@35369
   371
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
haftmann@35369
   372
    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
haftmann@35369
   373
  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
haftmann@35369
   374
    by (rule normalize_coprime) simp
haftmann@35369
   375
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
haftmann@35369
   376
  with quotient_of_unique have
haftmann@35369
   377
    "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
haftmann@35369
   378
    by (rule the1_equality)
haftmann@35369
   379
  then show ?thesis by (simp add: quotient_of_def)
haftmann@35369
   380
qed
haftmann@35369
   381
haftmann@35369
   382
lemma quotient_of_number [simp]:
haftmann@35369
   383
  "quotient_of 0 = (0, 1)"
haftmann@35369
   384
  "quotient_of 1 = (1, 1)"
haftmann@35369
   385
  "quotient_of (number_of k) = (number_of k, 1)"
haftmann@35369
   386
  by (simp_all add: rat_number_expand quotient_of_Fract)
nipkow@33805
   387
haftmann@35369
   388
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   389
  by (simp add: quotient_of_Fract normalize_eq)
haftmann@35369
   390
haftmann@35369
   391
lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   392
  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
haftmann@35369
   393
haftmann@35369
   394
lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   395
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
nipkow@33805
   396
haftmann@35369
   397
lemma quotient_of_inject:
haftmann@35369
   398
  assumes "quotient_of a = quotient_of b"
haftmann@35369
   399
  shows "a = b"
haftmann@35369
   400
proof -
haftmann@35369
   401
  obtain p q r s where a: "a = Fract p q"
haftmann@35369
   402
    and b: "b = Fract r s"
haftmann@35369
   403
    and "q > 0" and "s > 0" by (cases a, cases b)
haftmann@35369
   404
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
haftmann@35369
   405
qed
haftmann@35369
   406
haftmann@35369
   407
lemma quotient_of_inject_eq:
haftmann@35369
   408
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
haftmann@35369
   409
  by (auto simp add: quotient_of_inject)
nipkow@33805
   410
haftmann@27551
   411
haftmann@27551
   412
subsubsection {* The field of rational numbers *}
haftmann@27551
   413
haftmann@27551
   414
instantiation rat :: "{field, division_by_zero}"
haftmann@27551
   415
begin
haftmann@27551
   416
haftmann@27551
   417
definition
haftmann@35369
   418
  inverse_rat_def:
haftmann@27551
   419
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
haftmann@27551
   420
     ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
haftmann@27551
   421
haftmann@27652
   422
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
haftmann@27551
   423
proof -
haftmann@27551
   424
  have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
haftmann@27551
   425
    by (auto simp add: congruent_def mult_commute)
haftmann@27551
   426
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
huffman@27509
   427
qed
huffman@27509
   428
haftmann@27551
   429
definition
haftmann@35369
   430
  divide_rat_def: "q / r = q * inverse (r::rat)"
haftmann@27551
   431
haftmann@27652
   432
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
haftmann@27652
   433
  by (simp add: divide_rat_def)
haftmann@27551
   434
haftmann@27551
   435
instance proof
haftmann@27652
   436
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
haftmann@27551
   437
    (simp add: rat_number_collapse)
haftmann@27551
   438
next
haftmann@27551
   439
  fix q :: rat
haftmann@27551
   440
  assume "q \<noteq> 0"
haftmann@27551
   441
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
huffman@35216
   442
   (simp_all add: rat_number_expand eq_rat)
haftmann@27551
   443
next
haftmann@27551
   444
  fix q r :: rat
haftmann@27551
   445
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
haftmann@27551
   446
qed
haftmann@27551
   447
haftmann@27551
   448
end
haftmann@27551
   449
haftmann@27551
   450
haftmann@27551
   451
subsubsection {* Various *}
haftmann@27551
   452
haftmann@27551
   453
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
haftmann@27652
   454
  by (simp add: rat_number_expand)
haftmann@27551
   455
haftmann@27551
   456
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652
   457
  by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551
   458
haftmann@35369
   459
lemma Fract_number_of_quotient:
haftmann@27551
   460
  "Fract (number_of k) (number_of l) = number_of k / number_of l"
haftmann@27551
   461
  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
haftmann@27551
   462
haftmann@35369
   463
lemma Fract_1_number_of:
haftmann@27652
   464
  "Fract 1 (number_of k) = 1 / number_of k"
haftmann@27652
   465
  unfolding Fract_of_int_quotient number_of_eq by simp
haftmann@27551
   466
haftmann@27551
   467
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   468
huffman@27509
   469
instantiation rat :: linorder
huffman@27509
   470
begin
huffman@27509
   471
huffman@27509
   472
definition
haftmann@35369
   473
  le_rat_def:
huffman@27509
   474
   "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551
   475
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
haftmann@27551
   476
haftmann@27652
   477
lemma le_rat [simp]:
haftmann@27551
   478
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   479
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27551
   480
proof -
haftmann@27551
   481
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
haftmann@27551
   482
    respects2 ratrel"
haftmann@27551
   483
  proof (clarsimp simp add: congruent2_def)
haftmann@27551
   484
    fix a b a' b' c d c' d'::int
haftmann@27551
   485
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
haftmann@27551
   486
    assume eq1: "a * b' = a' * b"
haftmann@27551
   487
    assume eq2: "c * d' = c' * d"
haftmann@27551
   488
haftmann@27551
   489
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
haftmann@27551
   490
    {
haftmann@27551
   491
      fix a b c d x :: int assume x: "x \<noteq> 0"
haftmann@27551
   492
      have "?le a b c d = ?le (a * x) (b * x) c d"
haftmann@27551
   493
      proof -
haftmann@27551
   494
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
haftmann@27551
   495
        hence "?le a b c d =
haftmann@27551
   496
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
haftmann@27551
   497
          by (simp add: mult_le_cancel_right)
haftmann@27551
   498
        also have "... = ?le (a * x) (b * x) c d"
haftmann@27551
   499
          by (simp add: mult_ac)
haftmann@27551
   500
        finally show ?thesis .
haftmann@27551
   501
      qed
haftmann@27551
   502
    } note le_factor = this
haftmann@27551
   503
haftmann@27551
   504
    let ?D = "b * d" and ?D' = "b' * d'"
haftmann@27551
   505
    from neq have D: "?D \<noteq> 0" by simp
haftmann@27551
   506
    from neq have "?D' \<noteq> 0" by simp
haftmann@27551
   507
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
haftmann@27551
   508
      by (rule le_factor)
chaieb@27668
   509
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
haftmann@27551
   510
      by (simp add: mult_ac)
haftmann@27551
   511
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
haftmann@27551
   512
      by (simp only: eq1 eq2)
haftmann@27551
   513
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
haftmann@27551
   514
      by (simp add: mult_ac)
haftmann@27551
   515
    also from D have "... = ?le a' b' c' d'"
haftmann@27551
   516
      by (rule le_factor [symmetric])
haftmann@27551
   517
    finally show "?le a b c d = ?le a' b' c' d'" .
haftmann@27551
   518
  qed
haftmann@27551
   519
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
haftmann@27551
   520
qed
huffman@27509
   521
huffman@27509
   522
definition
haftmann@35369
   523
  less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509
   524
haftmann@27652
   525
lemma less_rat [simp]:
haftmann@27551
   526
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   527
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   528
  using assms by (simp add: less_rat_def eq_rat order_less_le)
huffman@27509
   529
huffman@27509
   530
instance proof
paulson@14365
   531
  fix q r s :: rat
paulson@14365
   532
  {
paulson@14365
   533
    assume "q \<le> r" and "r \<le> s"
haftmann@35369
   534
    then show "q \<le> s" 
haftmann@35369
   535
    proof (induct q, induct r, induct s)
paulson@14365
   536
      fix a b c d e f :: int
haftmann@35369
   537
      assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   538
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365
   539
      show "Fract a b \<le> Fract e f"
paulson@14365
   540
      proof -
paulson@14365
   541
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365
   542
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365
   543
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365
   544
        proof -
paulson@14365
   545
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   546
            by simp
paulson@14365
   547
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   548
        qed
chaieb@27668
   549
        also have "... = (c * f) * (d * f) * (b * b)" by algebra
paulson@14365
   550
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   551
        proof -
paulson@14365
   552
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
haftmann@27652
   553
            by simp
paulson@14365
   554
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   555
        qed
paulson@14365
   556
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   557
          by (simp only: mult_ac)
paulson@14365
   558
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   559
          by (simp add: mult_le_cancel_right)
haftmann@27652
   560
        with neq show ?thesis by simp
paulson@14365
   561
      qed
paulson@14365
   562
    qed
paulson@14365
   563
  next
paulson@14365
   564
    assume "q \<le> r" and "r \<le> q"
haftmann@35369
   565
    then show "q = r"
haftmann@35369
   566
    proof (induct q, induct r)
paulson@14365
   567
      fix a b c d :: int
haftmann@35369
   568
      assume neq: "b > 0"  "d > 0"
paulson@14365
   569
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   570
      show "Fract a b = Fract c d"
paulson@14365
   571
      proof -
paulson@14365
   572
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   573
          by simp
paulson@14365
   574
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   575
        proof -
paulson@14365
   576
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
haftmann@27652
   577
            by simp
paulson@14365
   578
          thus ?thesis by (simp only: mult_ac)
paulson@14365
   579
        qed
paulson@14365
   580
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   581
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   582
        ultimately have "a * d = c * b" by simp
paulson@14365
   583
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   584
      qed
paulson@14365
   585
    qed
paulson@14365
   586
  next
paulson@14365
   587
    show "q \<le> q"
haftmann@27652
   588
      by (induct q) simp
haftmann@27682
   589
    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
haftmann@27682
   590
      by (induct q, induct r) (auto simp add: le_less mult_commute)
paulson@14365
   591
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   592
      by (induct q, induct r)
haftmann@27652
   593
         (simp add: mult_commute, rule linorder_linear)
paulson@14365
   594
  }
paulson@14365
   595
qed
paulson@14365
   596
huffman@27509
   597
end
huffman@27509
   598
haftmann@27551
   599
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571
   600
begin
haftmann@25571
   601
haftmann@25571
   602
definition
haftmann@35369
   603
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551
   604
haftmann@27652
   605
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
huffman@35216
   606
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
haftmann@27551
   607
haftmann@27551
   608
definition
haftmann@35369
   609
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551
   610
haftmann@27652
   611
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551
   612
  unfolding Fract_of_int_eq
haftmann@27652
   613
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551
   614
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   615
haftmann@27551
   616
definition
haftmann@25571
   617
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571
   618
haftmann@25571
   619
definition
haftmann@25571
   620
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571
   621
haftmann@27551
   622
instance by intro_classes
haftmann@27551
   623
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456
   624
haftmann@25571
   625
end
haftmann@25571
   626
haftmann@35028
   627
instance rat :: linordered_field
haftmann@27551
   628
proof
paulson@14365
   629
  fix q r s :: rat
paulson@14365
   630
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   631
  proof (induct q, induct r, induct s)
paulson@14365
   632
    fix a b c d e f :: int
haftmann@35369
   633
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   634
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   635
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   636
    proof -
paulson@14365
   637
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   638
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   639
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   640
        by simp
paulson@14365
   641
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   642
        by (simp add: mult_le_cancel_right)
haftmann@27652
   643
      with neq show ?thesis by (simp add: mult_ac int_distrib)
paulson@14365
   644
    qed
paulson@14365
   645
  qed
paulson@14365
   646
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   647
  proof (induct q, induct r, induct s)
paulson@14365
   648
    fix a b c d e f :: int
haftmann@35369
   649
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   650
    assume le: "Fract a b < Fract c d"
paulson@14365
   651
    assume gt: "0 < Fract e f"
paulson@14365
   652
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   653
    proof -
paulson@14365
   654
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   655
      from neq gt have "0 < ?E"
haftmann@27652
   656
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
paulson@14365
   657
      moreover from neq have "0 < ?F"
paulson@14365
   658
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   659
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   660
        by simp
paulson@14365
   661
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   662
        by (simp add: mult_less_cancel_right)
paulson@14365
   663
      with neq show ?thesis
haftmann@27652
   664
        by (simp add: mult_ac)
paulson@14365
   665
    qed
paulson@14365
   666
  qed
haftmann@27551
   667
qed auto
paulson@14365
   668
haftmann@27551
   669
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   670
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   671
  shows "P q"
paulson@14365
   672
proof (cases q)
haftmann@27551
   673
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   674
  proof -
paulson@14365
   675
    fix a::int and b::int
paulson@14365
   676
    assume b: "b < 0"
paulson@14365
   677
    hence "0 < -b" by simp
paulson@14365
   678
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   679
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   680
  qed
paulson@14365
   681
  case (Fract a b)
paulson@14365
   682
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   683
qed
paulson@14365
   684
paulson@14365
   685
lemma zero_less_Fract_iff:
huffman@30095
   686
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095
   687
  by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095
   688
huffman@30095
   689
lemma Fract_less_zero_iff:
huffman@30095
   690
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095
   691
  by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095
   692
huffman@30095
   693
lemma zero_le_Fract_iff:
huffman@30095
   694
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095
   695
  by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095
   696
huffman@30095
   697
lemma Fract_le_zero_iff:
huffman@30095
   698
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095
   699
  by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095
   700
huffman@30095
   701
lemma one_less_Fract_iff:
huffman@30095
   702
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095
   703
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   704
huffman@30095
   705
lemma Fract_less_one_iff:
huffman@30095
   706
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095
   707
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   708
huffman@30095
   709
lemma one_le_Fract_iff:
huffman@30095
   710
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095
   711
  by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095
   712
huffman@30095
   713
lemma Fract_le_one_iff:
huffman@30095
   714
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095
   715
  by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365
   716
paulson@14378
   717
huffman@30097
   718
subsubsection {* Rationals are an Archimedean field *}
huffman@30097
   719
huffman@30097
   720
lemma rat_floor_lemma:
huffman@30097
   721
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097
   722
proof -
huffman@30097
   723
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@35293
   724
    by (cases "b = 0", simp, simp add: of_int_rat)
huffman@30097
   725
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@35293
   726
    unfolding Fract_of_int_quotient
huffman@35293
   727
    by (rule linorder_cases [of b 0])
huffman@35293
   728
       (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
huffman@30097
   729
  ultimately show ?thesis by simp
huffman@30097
   730
qed
huffman@30097
   731
huffman@30097
   732
instance rat :: archimedean_field
huffman@30097
   733
proof
huffman@30097
   734
  fix r :: rat
huffman@30097
   735
  show "\<exists>z. r \<le> of_int z"
huffman@30097
   736
  proof (induct r)
huffman@30097
   737
    case (Fract a b)
huffman@35293
   738
    have "Fract a b \<le> of_int (a div b + 1)"
huffman@35293
   739
      using rat_floor_lemma [of a b] by simp
huffman@30097
   740
    then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097
   741
  qed
huffman@30097
   742
qed
huffman@30097
   743
huffman@35293
   744
lemma floor_Fract: "floor (Fract a b) = a div b"
huffman@35293
   745
  using rat_floor_lemma [of a b]
huffman@30097
   746
  by (simp add: floor_unique)
huffman@30097
   747
huffman@30097
   748
haftmann@31100
   749
subsection {* Linear arithmetic setup *}
paulson@14387
   750
haftmann@31100
   751
declaration {*
haftmann@31100
   752
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100
   753
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100
   754
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100
   755
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100
   756
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100
   757
      @{thm True_implies_equals},
haftmann@31100
   758
      read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},
haftmann@31100
   759
      @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100
   760
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100
   761
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100
   762
      @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100
   763
      @{thm of_int_of_nat_eq}]
haftmann@31100
   764
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100
   765
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100
   766
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100
   767
*}
paulson@14387
   768
huffman@23342
   769
huffman@23342
   770
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   771
haftmann@24198
   772
class field_char_0 = field + ring_char_0
huffman@23342
   773
haftmann@35028
   774
subclass (in linordered_field) field_char_0 ..
huffman@23342
   775
haftmann@27551
   776
context field_char_0
haftmann@27551
   777
begin
haftmann@27551
   778
haftmann@27551
   779
definition of_rat :: "rat \<Rightarrow> 'a" where
haftmann@35369
   780
  "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
huffman@23342
   781
haftmann@27551
   782
end
haftmann@27551
   783
huffman@23342
   784
lemma of_rat_congruent:
haftmann@27551
   785
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
huffman@23342
   786
apply (rule congruent.intro)
huffman@23342
   787
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   788
apply (simp only: of_int_mult [symmetric])
huffman@23342
   789
done
huffman@23342
   790
haftmann@27551
   791
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
haftmann@27551
   792
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
huffman@23342
   793
huffman@23342
   794
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@23342
   795
by (simp add: Zero_rat_def of_rat_rat)
huffman@23342
   796
huffman@23342
   797
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@23342
   798
by (simp add: One_rat_def of_rat_rat)
huffman@23342
   799
huffman@23342
   800
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
haftmann@27652
   801
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
huffman@23342
   802
huffman@23343
   803
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
haftmann@27652
   804
by (induct a, simp add: of_rat_rat)
huffman@23343
   805
huffman@23343
   806
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   807
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   808
huffman@23342
   809
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
haftmann@27652
   810
apply (induct a, induct b, simp add: of_rat_rat)
huffman@23342
   811
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   812
done
huffman@23342
   813
huffman@23342
   814
lemma nonzero_of_rat_inverse:
huffman@23342
   815
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   816
apply (rule inverse_unique [symmetric])
huffman@23343
   817
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   818
done
huffman@23342
   819
huffman@23342
   820
lemma of_rat_inverse:
huffman@23342
   821
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
huffman@23342
   822
   inverse (of_rat a)"
huffman@23342
   823
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   824
huffman@23342
   825
lemma nonzero_of_rat_divide:
huffman@23342
   826
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   827
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   828
huffman@23342
   829
lemma of_rat_divide:
huffman@23342
   830
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
huffman@23342
   831
   = of_rat a / of_rat b"
haftmann@27652
   832
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342
   833
huffman@23343
   834
lemma of_rat_power:
haftmann@31017
   835
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273
   836
by (induct n) (simp_all add: of_rat_mult)
huffman@23343
   837
huffman@23343
   838
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@23343
   839
apply (induct a, induct b)
huffman@23343
   840
apply (simp add: of_rat_rat eq_rat)
huffman@23343
   841
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   842
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   843
done
huffman@23343
   844
haftmann@27652
   845
lemma of_rat_less:
haftmann@35028
   846
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652
   847
proof (induct r, induct s)
haftmann@27652
   848
  fix a b c d :: int
haftmann@27652
   849
  assume not_zero: "b > 0" "d > 0"
haftmann@27652
   850
  then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652
   851
  have of_int_divide_less_eq:
haftmann@27652
   852
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652
   853
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652
   854
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@35028
   855
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
haftmann@27652
   856
    \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652
   857
    using not_zero `b * d > 0`
haftmann@27652
   858
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652
   859
qed
haftmann@27652
   860
haftmann@27652
   861
lemma of_rat_less_eq:
haftmann@35028
   862
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652
   863
  unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652
   864
huffman@23343
   865
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   866
haftmann@27652
   867
lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343
   868
proof
huffman@23343
   869
  fix a
huffman@23343
   870
  show "of_rat a = id a"
huffman@23343
   871
  by (induct a)
haftmann@27652
   872
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343
   873
qed
huffman@23343
   874
huffman@23343
   875
text{*Collapse nested embeddings*}
huffman@23343
   876
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   877
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   878
huffman@23343
   879
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652
   880
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343
   881
huffman@23343
   882
lemma of_rat_number_of_eq [simp]:
huffman@23343
   883
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
huffman@23343
   884
by (simp add: number_of_eq)
huffman@23343
   885
haftmann@23879
   886
lemmas zero_rat = Zero_rat_def
haftmann@23879
   887
lemmas one_rat = One_rat_def
haftmann@23879
   888
haftmann@24198
   889
abbreviation
haftmann@24198
   890
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   891
where
haftmann@24198
   892
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   893
haftmann@24198
   894
abbreviation
haftmann@24198
   895
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   896
where
haftmann@24198
   897
  "rat_of_int \<equiv> of_int"
haftmann@24198
   898
huffman@28010
   899
subsection {* The Set of Rational Numbers *}
berghofe@24533
   900
nipkow@28001
   901
context field_char_0
nipkow@28001
   902
begin
nipkow@28001
   903
nipkow@28001
   904
definition
nipkow@28001
   905
  Rats  :: "'a set" where
haftmann@35369
   906
  "Rats = range of_rat"
nipkow@28001
   907
nipkow@28001
   908
notation (xsymbols)
nipkow@28001
   909
  Rats  ("\<rat>")
nipkow@28001
   910
nipkow@28001
   911
end
nipkow@28001
   912
huffman@28010
   913
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010
   914
by (simp add: Rats_def)
huffman@28010
   915
huffman@28010
   916
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010
   917
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010
   918
huffman@28010
   919
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010
   920
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010
   921
huffman@28010
   922
lemma Rats_number_of [simp]:
huffman@28010
   923
  "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
huffman@28010
   924
by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
huffman@28010
   925
huffman@28010
   926
lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010
   927
apply (unfold Rats_def)
huffman@28010
   928
apply (rule range_eqI)
huffman@28010
   929
apply (rule of_rat_0 [symmetric])
huffman@28010
   930
done
huffman@28010
   931
huffman@28010
   932
lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010
   933
apply (unfold Rats_def)
huffman@28010
   934
apply (rule range_eqI)
huffman@28010
   935
apply (rule of_rat_1 [symmetric])
huffman@28010
   936
done
huffman@28010
   937
huffman@28010
   938
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010
   939
apply (auto simp add: Rats_def)
huffman@28010
   940
apply (rule range_eqI)
huffman@28010
   941
apply (rule of_rat_add [symmetric])
huffman@28010
   942
done
huffman@28010
   943
huffman@28010
   944
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010
   945
apply (auto simp add: Rats_def)
huffman@28010
   946
apply (rule range_eqI)
huffman@28010
   947
apply (rule of_rat_minus [symmetric])
huffman@28010
   948
done
huffman@28010
   949
huffman@28010
   950
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010
   951
apply (auto simp add: Rats_def)
huffman@28010
   952
apply (rule range_eqI)
huffman@28010
   953
apply (rule of_rat_diff [symmetric])
huffman@28010
   954
done
huffman@28010
   955
huffman@28010
   956
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010
   957
apply (auto simp add: Rats_def)
huffman@28010
   958
apply (rule range_eqI)
huffman@28010
   959
apply (rule of_rat_mult [symmetric])
huffman@28010
   960
done
huffman@28010
   961
huffman@28010
   962
lemma nonzero_Rats_inverse:
huffman@28010
   963
  fixes a :: "'a::field_char_0"
huffman@28010
   964
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   965
apply (auto simp add: Rats_def)
huffman@28010
   966
apply (rule range_eqI)
huffman@28010
   967
apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010
   968
done
huffman@28010
   969
huffman@28010
   970
lemma Rats_inverse [simp]:
huffman@28010
   971
  fixes a :: "'a::{field_char_0,division_by_zero}"
huffman@28010
   972
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   973
apply (auto simp add: Rats_def)
huffman@28010
   974
apply (rule range_eqI)
huffman@28010
   975
apply (rule of_rat_inverse [symmetric])
huffman@28010
   976
done
huffman@28010
   977
huffman@28010
   978
lemma nonzero_Rats_divide:
huffman@28010
   979
  fixes a b :: "'a::field_char_0"
huffman@28010
   980
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   981
apply (auto simp add: Rats_def)
huffman@28010
   982
apply (rule range_eqI)
huffman@28010
   983
apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010
   984
done
huffman@28010
   985
huffman@28010
   986
lemma Rats_divide [simp]:
huffman@28010
   987
  fixes a b :: "'a::{field_char_0,division_by_zero}"
huffman@28010
   988
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   989
apply (auto simp add: Rats_def)
huffman@28010
   990
apply (rule range_eqI)
huffman@28010
   991
apply (rule of_rat_divide [symmetric])
huffman@28010
   992
done
huffman@28010
   993
huffman@28010
   994
lemma Rats_power [simp]:
haftmann@31017
   995
  fixes a :: "'a::field_char_0"
huffman@28010
   996
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010
   997
apply (auto simp add: Rats_def)
huffman@28010
   998
apply (rule range_eqI)
huffman@28010
   999
apply (rule of_rat_power [symmetric])
huffman@28010
  1000
done
huffman@28010
  1001
huffman@28010
  1002
lemma Rats_cases [cases set: Rats]:
huffman@28010
  1003
  assumes "q \<in> \<rat>"
huffman@28010
  1004
  obtains (of_rat) r where "q = of_rat r"
huffman@28010
  1005
  unfolding Rats_def
huffman@28010
  1006
proof -
huffman@28010
  1007
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010
  1008
  then obtain r where "q = of_rat r" ..
huffman@28010
  1009
  then show thesis ..
huffman@28010
  1010
qed
huffman@28010
  1011
huffman@28010
  1012
lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010
  1013
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010
  1014
  by (rule Rats_cases) auto
huffman@28010
  1015
nipkow@28001
  1016
berghofe@24533
  1017
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
  1018
haftmann@35369
  1019
definition Frct :: "int \<times> int \<Rightarrow> rat" where
haftmann@35369
  1020
  [simp]: "Frct p = Fract (fst p) (snd p)"
haftmann@35369
  1021
haftmann@35369
  1022
code_abstype Frct quotient_of
haftmann@35369
  1023
proof (rule eq_reflection)
haftmann@35373
  1024
  fix r :: rat
haftmann@35373
  1025
  show "Frct (quotient_of r) = r" by (cases r) (auto intro: quotient_of_eq)
haftmann@35369
  1026
qed
haftmann@35369
  1027
haftmann@35369
  1028
lemma Frct_code_post [code_post]:
haftmann@35369
  1029
  "Frct (0, k) = 0"
haftmann@35369
  1030
  "Frct (k, 0) = 0"
haftmann@35369
  1031
  "Frct (1, 1) = 1"
haftmann@35369
  1032
  "Frct (number_of k, 1) = number_of k"
haftmann@35369
  1033
  "Frct (1, number_of k) = 1 / number_of k"
haftmann@35369
  1034
  "Frct (number_of k, number_of l) = number_of k / number_of l"
haftmann@35369
  1035
  by (simp_all add: rat_number_collapse Fract_number_of_quotient Fract_1_number_of)
haftmann@35369
  1036
haftmann@35369
  1037
declare quotient_of_Fract [code abstract]
haftmann@35369
  1038
haftmann@35369
  1039
lemma rat_zero_code [code abstract]:
haftmann@35369
  1040
  "quotient_of 0 = (0, 1)"
haftmann@35369
  1041
  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1042
haftmann@35369
  1043
lemma rat_one_code [code abstract]:
haftmann@35369
  1044
  "quotient_of 1 = (1, 1)"
haftmann@35369
  1045
  by (simp add: One_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1046
haftmann@35369
  1047
lemma rat_plus_code [code abstract]:
haftmann@35369
  1048
  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1049
     in normalize (a * d + b * c, c * d))"
haftmann@35369
  1050
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@27652
  1051
haftmann@35369
  1052
lemma rat_uminus_code [code abstract]:
haftmann@35369
  1053
  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
haftmann@35369
  1054
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1055
haftmann@35369
  1056
lemma rat_minus_code [code abstract]:
haftmann@35369
  1057
  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1058
     in normalize (a * d - b * c, c * d))"
haftmann@35369
  1059
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1060
haftmann@35369
  1061
lemma rat_times_code [code abstract]:
haftmann@35369
  1062
  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1063
     in normalize (a * b, c * d))"
haftmann@35369
  1064
  by (cases p, cases q) (simp add: quotient_of_Fract)
berghofe@24533
  1065
haftmann@35369
  1066
lemma rat_inverse_code [code abstract]:
haftmann@35369
  1067
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
haftmann@35369
  1068
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
haftmann@35369
  1069
proof (cases p)
haftmann@35369
  1070
  case (Fract a b) then show ?thesis
haftmann@35369
  1071
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
haftmann@35369
  1072
qed
haftmann@35369
  1073
haftmann@35369
  1074
lemma rat_divide_code [code abstract]:
haftmann@35369
  1075
  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1076
     in normalize (a * d, c * b))"
haftmann@35369
  1077
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1078
haftmann@35369
  1079
lemma rat_abs_code [code abstract]:
haftmann@35369
  1080
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
haftmann@35369
  1081
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1082
haftmann@35369
  1083
lemma rat_sgn_code [code abstract]:
haftmann@35369
  1084
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
haftmann@35369
  1085
proof (cases p)
haftmann@35369
  1086
  case (Fract a b) then show ?thesis
haftmann@35369
  1087
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
haftmann@35369
  1088
qed
berghofe@24533
  1089
haftmann@26513
  1090
instantiation rat :: eq
haftmann@26513
  1091
begin
haftmann@26513
  1092
haftmann@35369
  1093
definition [code]:
haftmann@35369
  1094
  "eq_class.eq a b \<longleftrightarrow> quotient_of a = quotient_of b"
haftmann@26513
  1095
haftmann@35369
  1096
instance proof
haftmann@35369
  1097
qed (simp add: eq_rat_def quotient_of_inject_eq)
haftmann@26513
  1098
haftmann@28351
  1099
lemma rat_eq_refl [code nbe]:
haftmann@28351
  1100
  "eq_class.eq (r::rat) r \<longleftrightarrow> True"
haftmann@28351
  1101
  by (rule HOL.eq_refl)
haftmann@28351
  1102
haftmann@26513
  1103
end
berghofe@24533
  1104
haftmann@35369
  1105
lemma rat_less_eq_code [code]:
haftmann@35369
  1106
  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
haftmann@35726
  1107
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1108
haftmann@35369
  1109
lemma rat_less_code [code]:
haftmann@35369
  1110
  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
haftmann@35726
  1111
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1112
haftmann@35369
  1113
lemma [code]:
haftmann@35369
  1114
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
haftmann@35369
  1115
  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
haftmann@27652
  1116
haftmann@31203
  1117
definition (in term_syntax)
haftmann@32657
  1118
  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
haftmann@32657
  1119
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203
  1120
haftmann@31203
  1121
notation fcomp (infixl "o>" 60)
haftmann@31203
  1122
notation scomp (infixl "o\<rightarrow>" 60)
haftmann@31203
  1123
haftmann@31203
  1124
instantiation rat :: random
haftmann@31203
  1125
begin
haftmann@31203
  1126
haftmann@31203
  1127
definition
haftmann@31641
  1128
  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205
  1129
     let j = Code_Numeral.int_of (denom + 1)
haftmann@32657
  1130
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203
  1131
haftmann@31203
  1132
instance ..
haftmann@31203
  1133
haftmann@31203
  1134
end
haftmann@31203
  1135
haftmann@31203
  1136
no_notation fcomp (infixl "o>" 60)
haftmann@31203
  1137
no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@31203
  1138
haftmann@24622
  1139
text {* Setup for SML code generator *}
berghofe@24533
  1140
berghofe@24533
  1141
types_code
berghofe@24533
  1142
  rat ("(int */ int)")
berghofe@24533
  1143
attach (term_of) {*
berghofe@24533
  1144
fun term_of_rat (p, q) =
haftmann@24622
  1145
  let
haftmann@35372
  1146
    val rT = Type ("Rat.rat", [])
berghofe@24533
  1147
  in
berghofe@24533
  1148
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
berghofe@25885
  1149
    else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
berghofe@24533
  1150
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24533
  1151
  end;
berghofe@24533
  1152
*}
berghofe@24533
  1153
attach (test) {*
berghofe@24533
  1154
fun gen_rat i =
berghofe@24533
  1155
  let
berghofe@24533
  1156
    val p = random_range 0 i;
berghofe@24533
  1157
    val q = random_range 1 (i + 1);
berghofe@24533
  1158
    val g = Integer.gcd p q;
wenzelm@24630
  1159
    val p' = p div g;
wenzelm@24630
  1160
    val q' = q div g;
berghofe@25885
  1161
    val r = (if one_of [true, false] then p' else ~ p',
haftmann@31666
  1162
      if p' = 0 then 1 else q')
berghofe@24533
  1163
  in
berghofe@25885
  1164
    (r, fn () => term_of_rat r)
berghofe@24533
  1165
  end;
berghofe@24533
  1166
*}
berghofe@24533
  1167
berghofe@24533
  1168
consts_code
haftmann@27551
  1169
  Fract ("(_,/ _)")
berghofe@24533
  1170
berghofe@24533
  1171
consts_code
haftmann@35375
  1172
  quotient_of ("{*normalize*}")
haftmann@35375
  1173
haftmann@35375
  1174
consts_code
berghofe@24533
  1175
  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
berghofe@24533
  1176
attach {*
haftmann@31674
  1177
fun rat_of_int i = (i, 1);
berghofe@24533
  1178
*}
berghofe@24533
  1179
blanchet@33197
  1180
setup {*
wenzelm@33209
  1181
  Nitpick.register_frac_type @{type_name rat}
wenzelm@33209
  1182
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209
  1183
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209
  1184
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209
  1185
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209
  1186
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209
  1187
    (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
wenzelm@33209
  1188
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
wenzelm@33209
  1189
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
wenzelm@33209
  1190
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac}),
wenzelm@35402
  1191
    (@{const_name field_char_0_class.Rats}, @{const_abbrev UNIV})]
blanchet@33197
  1192
*}
blanchet@33197
  1193
blanchet@33197
  1194
lemmas [nitpick_def] = inverse_rat_inst.inverse_rat
wenzelm@33209
  1195
  number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_eq_rat
wenzelm@33209
  1196
  plus_rat_inst.plus_rat times_rat_inst.times_rat uminus_rat_inst.uminus_rat
wenzelm@33209
  1197
  zero_rat_inst.zero_rat
blanchet@33197
  1198
huffman@35343
  1199
subsection{* Float syntax *}
huffman@35343
  1200
huffman@35343
  1201
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
huffman@35343
  1202
huffman@35343
  1203
use "Tools/float_syntax.ML"
huffman@35343
  1204
setup Float_Syntax.setup
huffman@35343
  1205
huffman@35343
  1206
text{* Test: *}
huffman@35343
  1207
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
huffman@35343
  1208
by simp
huffman@35343
  1209
huffman@29880
  1210
end